Proporional Reasoning Focus on Afer his lesson, you will be able o... solve problems using proporional reasoning use more han one mehod o solve proporional reasoning problems When you go snowboarding or skiing, you use proporional reasoning o deermine he correc lengh of ski or board for you o use. This involves using he raio of your heigh o he lengh of he ski or board. To deermine he correc widh of he board so ha your fee do no hang over, you use he raio of he wais of he board, which is he narrowes par, o your boo size. Riders wih small fee need narrower boards han riders wih big fee. When you draw a porrai in ar class, you use proporional reasoning o figure ou how large o make each facial feaure in relaion o he oher feaures and how o align he feaures on he head. Think abou where you have made comparisons. Where migh you have used proporional reasoning? compuer access shor sory sopwach sicky noes You can use a word processing program o coun he words for you. How do you use proporional reasoning? 1. Copy he following able ino your noebook. Pu wo rows under he column headings o record daa for wo ypiss. Suden Number of Words Time (min) 2. Work wih a parner. Selec a shor sory o ype. 4 4 2.3 Proporional Reasoning MHR 63
Typing performance involves boh speed and accuracy. Many compuer programs provide informaion abou words per minue, number of keysrokes, and error raes. 3. Use a sopwach and ake urns o ime each oher s yping. The imer ells he ypis when o begin and when o sop (afer 4 min). 4. a) Type a a comforable rae so you can avoid making many errors. b) Afer he ime is up, mark your sopping place in he ex using a sicky noe. c) Coun and record he oal number of words yped in 4 min. 5. Trade roles and repea sep 4 o ge daa for your parner. 6. a) Wha is he four-minue yping rae for each ypis? b) Wha is he uni rae for each ypis? proporion a relaionship ha says ha wo raios or wo raes are equal can be wrien in fracion form: 2 3 = 6 9 2 km 3 h = 6 km 9 h Wha number do you divide he denominaor by? Wha number do you divide he numeraor by? Are boh numbers he same? Why? Reflec on Your Findings 7. a) If each ypis coninued yping a he same rae, how many words could each person ype in 1 h? Approximaely how many pages is ha? b) Wha oher facors migh affec how long i akes o ype he enire sory? Give an esimae of he ime needed for each ypis o ype he sory. c) How did you use a proporion o find your answer o par a)? Example 1: Solve a Rae Problem Using Proporional Reasoning Elecriciy coss 11.58 for 2 kwh. How much does 30 kwh cos? Give your answer o he neares cen. Soluion Mehod 1: Use a Uni Rae A cos of 11.58 for 2 kwh can be expressed as he rae 11.58 2 kwh. Deermine he uni rae. 11.58 2 kwh = 5.79 1 kwh Elecriciy coss 5.79 per kwh or 5.79 /kwh. 30 kwh coss 30 5.79 = 173.7 C 30 5.79 = 173.7 So, 30 kwh coss 174 or $1.74 rounded o he neares cen. 173.7 means 173 and seven enhs cens. Recall ha five enhs or more is rounded o he nex cen. 64 MHR Chaper 2
Mehod 2: Use a Proporion Make a proporion o show wha you wan o find. 11.58 2 kwh = 30 kwh Solve he proporion. 11.58 2 kwh = 30 kwh 11.58 15 = 173.7 C 11.58 15 = 173.7 So, 30 kwh coss abou 174 or $1.74. Wha number do you muliply he denominaor by? Wha number do you muliply he numeraor by? 11.58 is abou 12. 12 15 = (10 15 + 2 15) = 150 + 30 = 180 The answer will be abou 180 or $1.80. There are 72 players on 8 baseball eams. Deermine he number of players on 2 eams. Show how o find he answer more han one way. Example 2: Solve a Raio Problem Using Proporional Reasoning A wildlife biologis wans o know how many rou are in a slough in Saskachewan. He capures and ags 24 rou and releases hem back ino he slough. Two weeks laer he reurns and capures 30 rou and finds ha 5 of hem are agged. He uses he following raios o esimae he number of fish in he slough: fish recapured wih ags = fish caugh and agged oal fish recapured oal fish in slough How many rou does he esimae are in he slough? Soluion Mehod 1: Use a Proporion in Lowes Terms fish recapured wih ags = fish agged oal fish recapured oal fish in slough 5 30 = 24 Se up he proporion using equal raios. 1 6 = 24 24 Reduce 5 30 o 1 6. Lieracy Link In Wesern Canada, a slough is a small lake or pond formed by rain or meled snow. Wildlife biologiss can show ha hese raios are equal if he fish populaion has an opporuniy o mix before he recapure. Sraegies Use a Variable 1 6 = 24 Make equivalen raios. 24 = 6 24 = 144 The biologis esimaes here are 144 rou in he slough. 2.3 Proporional Reasoning MHR 65
Mehod 2: Use he Original Proporion How would you find he soluion if you did no wrie 5 in lowes erms? 30 5 30 = 24 Divide o find wha number 4.8 5 30 = 24 4.8 = 30 4.8 = 144 muliplied by 5 gives 24. 24 5 = 4.8 Make equivalen raios. The biologis esimaes here are 144 rou in he slough. How much will a dozen erasers cos if hree erasers cos 75? Show how o find he answer in more han one way. A proporion is a relaionship ha says ha wo raios or wo raes are equal. A proporion can be expressed in fracion form: 5 3 1 2 = 5 10 60 si-ups = 20 si-ups 3 min 1 min 5 3 You can solve proporional reasoning problems using several differen mehods. A poao farmer can plan hree poao plans per 0.5 m 2. How many poao plans can she plan in an area of 85 m 2? 3 plans Use a uni rae. 0.5 m = 6 plans The uni rae is 6 plans/m 2. 2 1 m 2 Use a proporion. 170 6 85 = 510 The farmer can plan 510 poao plans. 3 plans 0.5 m = 2 85 m 2 170 Missing value is 3 170 = 510 The farmer can plan 510 poao plans. 66 MHR Chaper 2
1. Explain he similariies and differences beween a raio, a rae, and a proporion. Give an example of each one. 2. Your friend missed he lesson on proporions. Explain how o use a proporion o solve his problem. Cheryl is selling marbles. Wha is he cos of seven marbles? 3. a) Wrie a proporion based on he following scenario: Three balls cos $1.25. Wha is he cos of 12 balls? b) Solve he proporion. Marbles 2 3 4 Cos ( ) 70 105 140 For help wih #4 o #9, refer o Example 1 on pages 64 65. 4. Deermine he uni rae. a) Three dinner rolls cos 99. b) Seven idenical objecs have a mass of 14 kg. 5. Wha is he uni rae in each? a) Two pens cos 94. b) Four blocks sacked one on op of he oher are 24 cm high. 6. Delia was paid $35 for 5 h of babysiing. How much should she receive for 3 h? Use a uni rae o find he answer. 7. Solve #6 using a proporion. Show how o find he answer more han one way. 8. Deermine he missing value. a) 2 3 = 15 c) 30 45 = 6 b) 5 = 14 35 3 d) = 12 36 9. Deermine he missing value o make each rae equivalen. Include he unis. 60 km a) 3 h = b) $3 6 h 4 cans = $15 178 beas c) = d) 48 km = 192 km 2 min 1 min $16 For help wih #10 o #14, refer o Example 2 on pages 65 66. 10. Se up a proporion for each siuaion. a) If 10 beans have a mass of 17 g, hen 30 beans have a mass of 51 g. b) There are 13 boys for 15 girls in every classroom a Albany Middle School. If here are 65 boys in he school, hen here are 75 girls. c) On a map, 1 cm represens 25 km. Kendra wans o ride her bike 160 km. This disance is 6.4 cm on he map. 11. A small gear urns 18 imes in he same ime ha a large gear urns 4 imes. How many imes will he large gear urn if he small gear urns 54 imes? Draw a diagram o help se up a proporion and solve he problem. 2.3 Proporional Reasoning MHR 67
12. Se up a proporion for each siuaion using a variable. Do no find he answer. a) Waler makes his own oil and vinegar dressing. His recipe calls for 175 ml of olive oil and 50 ml of vinegar. Wha amoun of vinegar does he need o mix wih 300 ml of olive oil? b) A baseball player has a home run o srikeous raio of 3 : 17. How many home runs should he hi if he srikes ou 187 imes? 13. Two quarers have he same value as en nickels. Wha is he value of five quarers in nickels? 14. Las nigh 30 cm of snow fell in 6 h. If i coninues snowing a he same rae, how long will i ake for 45 cm of snow o fall? Deermine he answer wo differen ways. 15. Look a he paern. Se up a proporion you could use o find he number of small squares in Figure 7. Figure 1 Figure 2 Figure 3 Figure 4 16. A gardener akes a half hour o mow and weed a lawn ha measures 20 m by 15 m. He charges $25 per hour. How much should he gardener receive for a lawn ha measures 40 m by 30 m? 17. Fresh pickerel is 2 kg adverised in a local marke. a) How much will 6 kg of pickerel cos? fresh pickerel $17.60 1 kg = 1000 g b) Use a proporion o find he cos of 1600 g of pickerel. 18. A an amusemen park, a new hrill ride was inroduced. I coss $7.50 for 3 rides on he Wild Slider. a) Wha is he Wild Slider s uni rae per ride? b) A his rae, wha would i cos for 18 rides on Wild Slider? Deermine he answer wo differen ways. 19. Deermine he missing value in each equivalen fracion. a) 3 = 18 24 = 12 48 km b) = 144 km $16 = $64 20. A breakfas cereal conains corn, whea, and rice in he raio of 3 o 4 o 2. If a box of cereal conains 225 g of corn, how much rice does i conain? 21. David can saw a log ino hree pieces in 7 min. If he coninues sawing a a consan rae, how long will i ake him o saw a similar log ino six pieces? 22. The heigh of an objec compared o he lengh of is shadow is consan for all objecs a any given ime. ree heigh lengh of shadow = suden heigh lengh of shadow shadow shadow Use his informaion o help answer he following quesions. a) If a 15-m ree cass a 9-m shadow, wha is he heigh of a suden who cass a 1.08-m shadow? b) If a 50-m ower has a shadow 16 m long, how long is he shadow of a suden who is 1.5 m all? Give your answer o he neares cenimere. 68 MHR Chaper 2
23. According o he Guinness Book of World Records, he world s smalles horse is Thumbelina. Thumbelina is 42.5 cm all and eas abou 0.3 kg of food per day. A former world record holder ae food in he same proporion o is heigh. If i was 46.25 cm all, how much did i ea? Give your answer o he neares hundredh of a kilogram. 24. a) Describe a paern you could use o find he nex fracion in he following se of fracions. 1 2, 2 4, 3 6 b) Describe a paern you could use o find he nex fracion in he following se of fracions. 5 6, 10 12, 15 18 c) Choose any pair of fracions from par a) or par b). Muliply he numeraor of one fracion by he denominaor of he oher fracion. Repea for wo oher fracions in he same se. Wha do you noice abou he wo producs? d) Wha predicion could you make abou he cross-producs of any pair of equivalen fracions? Tes your predicion on anoher pair of equivalen fracions. 25. Mark esimaes ha frogs ea six insecs per hour and ha dragonflies ea nine insecs per hour. Assume a frog ress for 8 h each day and a dragonfly ress for 13 h each day. Neiher eas while resing. a) Deermine he daily rae of insecs eaen by a frog and a dragonfly. Which one eas more insecs per day? b) How many insecs would a dragonfly ea in a week? c) How many insecs would a frog ea in Augus? 26. Two circles have radii wih a raio of 1 o 2. Use a diagram o help answer he following quesions. a) Wha is he raio of heir circumferences? b) Wha is he raio of heir areas? 27. If a : b = 4 : 5, find he raio of 5a : 7b. 28. The dosage of a cerain medicine for a child is 2.5 ml for each 3 kg mass of he child. Wha is he dose, in millilires, for a child wih a mass of 16.5 kg? MATH LINK A horiaiki Greek salad has omaoes, cucumbers, fea cheese, and olives. I does no conain any leuce. Many culures have similar salads. For example, ezme salaasi is a Turkish omao-cucumber salad wih red peppers and paprika, bu wihou he fea and olives. a) I coss $7.60 o make he horiaiki salad for 12 people. Wha is he uni price? b) Choose and wrie down a recipe for a soup, a salad, or an appeizer ha serves beween 4 and 6 people. Record how much of each ingredien you will need o serve 10 people a your inernaional meal. 2.3 Proporional Reasoning MHR 69